I had some trouble with the police while I was in the U.S. on a visa, and I got deported based on an aggravated felony. They told me I am not supposed to try to return to the United States for my whole life. But I really need to go back, to take care of some personal things. What would happen if I crossed the border to the U.S. and caught caught by the Immigration people?
This is a dangerous idea. As you mentioned, any noncitizen who has been convicted of an aggravated felony and is subsequently deported from the United States is permanently inadmissible to the U.S. – that is, barred from returning even if the person found some new basis upon which to apply for a U.S. visa or green card. While many other types of inadmissible applicants qualify for waivers allowing them to go ahead with a new application despite being inadmissible, this is an unlikely – though not impossible – prospect for someone with an aggravated felony conviction.
Such a person who illegally reenters the U.S. after deportation can be charged with a federal crime. You would probably face a trial in federal court (but at least, unlike in immigration proceedings, you would be entitled to free legal representation if you cannot afford a lawyer.)
If you are convicted, the crime of reentry after deportation is punishable by monetary fines and prison sentences of up to 20 years. This is an especially long sentence due to the aggravated felony conviction. The relevant law on this is 8 U.S.C. § 1326(b)(2), I.N.A. § 276.
Consult an experienced immigration attorney for advice on what to do about your situation.